Whakaoti i te x^2-1,25x-3,75=0

$$x ^ { 2 } - 1,25 x - 3,75 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{265} + 5}{8} \approx 2.659852575$$

$$x=\frac{5-\sqrt{265}}{8}\approx -1.409852575$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$x^{2}-1.25x-3.75=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1.25 mō b, me -3.75 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1.25\right)±\sqrt{\left(-1.25\right)^{2}-4\left(-3.75\right)}}{2}$$

Pūruatia -1.25 mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-1.25\right)±\sqrt{1.5625-4\left(-3.75\right)}}{2}$$

Whakareatia -4 ki te -3.75.

$$x=\frac{-\left(-1.25\right)±\sqrt{1.5625+15}}{2}$$

Tāpiri 1.5625 ki te 15.

$$x=\frac{-\left(-1.25\right)±\sqrt{16.5625}}{2}$$

Tuhia te pūtakerua o te 16.5625.

$$x=\frac{-\left(-1.25\right)±\frac{\sqrt{265}}{4}}{2}$$

Ko te tauaro o -1.25 ko 1.25.

$$x=\frac{1.25±\frac{\sqrt{265}}{4}}{2}$$

Nā, me whakaoti te whārite x=\frac{1.25±\frac{\sqrt{265}}{4}}{2} ina he tāpiri te ±. Tāpiri 1.25 ki te \frac{\sqrt{265}}{4}.

$$x=\frac{\sqrt{265}+5}{2\times 4}$$

Whakawehe \frac{5+\sqrt{265}}{4} ki te 2.

$$x=\frac{\sqrt{265}+5}{8}$$

Nā, me whakaoti te whārite x=\frac{1.25±\frac{\sqrt{265}}{4}}{2} ina he tango te ±. Tango \frac{\sqrt{265}}{4} mai i 1.25.

$$x=\frac{5-\sqrt{265}}{2\times 4}$$

Whakawehe \frac{5-\sqrt{265}}{4} ki te 2.

$$x=\frac{5-\sqrt{265}}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{265}+5}{8} x=\frac{5-\sqrt{265}}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$x^{2}-1.25x-3.75=0$$

Me tāpiri 3.75 ki ngā taha e rua o te whārite.

$$x^{2}-1.25x-3.75-\left(-3.75\right)=-\left(-3.75\right)$$

Mā te tango i te -3.75 i a ia ake anō ka toe ko te 0.

$$x^{2}-1.25x=-\left(-3.75\right)$$

Tango -3.75 mai i 0.

$$x^{2}-1.25x=3.75$$

Whakawehea te -1.25, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -0.625. Nā, tāpiria te pūrua o te -0.625 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-1.25x+\left(-0.625\right)^{2}=3.75+\left(-0.625\right)^{2}$$

Pūruatia -0.625 mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-1.25x+0.390625=3.75+0.390625$$

Tāpiri 3.75 ki te 0.390625 mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-1.25x+0.390625=4.140625$$

Tauwehea x^{2}-1.25x+0.390625. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.625\right)^{2}=4.140625$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-0.625\right)^{2}}=\sqrt{4.140625}$$

Whakarūnātia.

$$x-0.625=\frac{\sqrt{265}}{8} x-0.625=-\frac{\sqrt{265}}{8}$$

Me tāpiri 0.625 ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{265}+5}{8} x=\frac{5-\sqrt{265}}{8}$$